I'm trying to rationalize whether or not it's possible whether we can manipulate the temperature of the earths mantle and in turn manipulate the movement of the tectonic plates in such a fashion that a man made mountain would form.

The rising and expanding temperature of the earths core pushes heat outwards towards the surface, moving the liquid inner state to the plastic outer state, where it then hardens and sinks back down to the liquid state. This circular rotation causes so much energy that it actually causes the outer most area our tectonic plates to shift. So what if we were to stimulate the intense heat in the mantle so that the heat continues to rise pushing the tectonic plates outwards crashing into each other in such an abrupt way that a mountain could form within in the course of decades or centuries rather than millennia. The simplest means I can come up with to stimulate the heat would be intense thermonuclear heat, from say a bomb or reactor, lowered past the crust into the mantle in a specific area where the mantle already has exceeding temperatures, say the ring of fire.

My question is, if the idea is even possible to begin with, how could we maintain such a massive amount of energy, and any adverse effects this could have on the planet if any at all.

A lot would depend on the depth of the crust you are attempting to change, the geology and geochemistry of the crust, and how intense the convection currents of the mantle are. To be honest the easier methodology would be to choose a very thin section of crust and do controlled bores to release magma. That is how volcanoes form. Mountains such as the Himalayas and the Alps are do not from in the manner you are looks at, so ones like that are out of the question anyways.

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I'm trying to rationalize whether or not it's possible whether we can manipulate the temperature of the earths mantle and in turn manipulate the movement of the tectonic plates in such a fashion that a man made mountain would form.

The rising and expanding temperature of the earths core pushes heat outwards towards the surface, moving the liquid inner state to the plastic outer state, where it then hardens and sinks back down to the liquid state. This circular rotation causes so much energy that it actually causes the outer most area our tectonic plates to shift. So what if we were to stimulate the intense heat in the mantle so that the heat continues to rise pushing the tectonic plates outwards crashing into each other in such an abrupt way that a mountain could form within in the course of decades or centuries rather than millennia. The simplest means I can come up with to stimulate the heat would be intense thermonuclear heat, from say a bomb or reactor, lowered past the crust into the mantle in a specific area where the mantle already has exceeding temperatures, say the ring of fire.

My question is, if the idea is even possible to begin with, how could we maintain such a massive amount of energy, and any adverse effects this could have on the planet if any at all.

No, I can't see how this could ever be feasible. As far as artificial stimulation of mantle convection goes, your problem is that the mantle is not liquid. It is a slightly plastic solid. So even if you raised the temperature at a rising limb of a convection cell, nothing would move much faster. You would still need millions of years.

However, as Palaeoichneum says, one might just conceivably be able to help form a new volcano. Even that would be very tricky. Volcanism depends on rising "diapirs" of magma: there isn't just magma everywhere below the surface, if you go down far enough. Rock only liquefies when something happens to lower the melting point, such as entrained water in a subducting slab of crust at a destructive plate margin, or else elevate the temperature, as at a mid-ocean ridge. So you'd have to find one of these diapirs and make a borehole down to it, or something.

My question is, if the idea is even possible to begin with, how could we maintain such a massive amount of energy, and any adverse effects this could have on the planet if any at all.

I suppose that idea (using nucleair fission to make an artifical mountain) is as dumb or as intelligent as putting nucleair waste in barrels, make a scientific calculation on how safe that solution (the barrels) will be over time and at the same time put the most radioactive waste on top of a missile